Symmetric Potentials

Can anyone help me to understand the following please? If a potential is symmetric does this mean that the eigenstates are either symmetric or antisymmetric? Is the ground state always symmetric and the first excited state always antisymmetric?

If a potential is symmetric does this mean that the eigenstates are either symmetric or antisymmetric?

Not necessarily, IMO. You can always choose the basis consisting of symmetric and antisymmetric eigenstates of the Hamiltonian though, as the parity operator commutes with the Hamiltonian. But this is not the same as what you ask, as a symmetric and an asymmetric eigenstates can have the same eigenvalue, so their linear combination will also be an eigenstate.

I guess this statement should be qualified somehow, because it is clearly doubtful in the case of a constant potential. My guess is the statement does not hold water in a more general case either, at least for the continuous spectrum. If you have infinitely high walls, I don't know, maybe you're right.